Theorems · Theorem · global analysis

exists_continuousLinearEquiv_fderiv_symm_eq

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace E]
  {f : E → F} {x : E},
  ContDiffAt 𝕜 2 f x →
    (fderiv 𝕜 f x).IsInvertible →
      ∃ N,
        ContDiffAt 𝕜 1 (fun y => ↑(N y)) x ∧
          ContDiffAt 𝕜 1 (fun y => ↑(N y).symm) x ∧
            (∀ᶠ (y : E) in nhds x, ↑(N y) = fderiv 𝕜 f y) ∧
              ∀ (v : E),
                (fderiv 𝕜 (fun y => ↑(N y).symm) x) v = -↑(N x).symm ∘SL (fderiv 𝕜 (fderiv 𝕜 f) x) v ∘SL ↑(N x).symm

If a C^2 map has an invertible derivative at a point, then nearby derivatives can be written as continuous linear equivs, which depend in a C^1 way on the point, as well as their inverse, and moreover one can compute the derivative of the inverse.

Defined in

Mathlib.Analysis.Calculus.VectorField

Cited by

0 results in Mathlib

Foundations

Depth 207 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace